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3 FACES FOR TWO QUBIT SEPARABLE STATES AND THE CONVEX HULLS OF TRIGONOMETRIC MOMENT CURVES
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Orbitopes
, 2009
"... An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. These highly symmetric convex bodies lie at the crossroads of several fields including convex geometry, algebraic geometry, and optimization. We present a selfcontained theory of orbitopes with particu ..."
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An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. These highly symmetric convex bodies lie at the crossroads of several fields including convex geometry, algebraic geometry, and optimization. We present a selfcontained theory of orbitopes with particular emphasis on instances arising from the groups SO(n) and O(n). These include SchurHorn orbitopes, tautological orbitopes, Carathéodory orbitopes, Veronese orbitopes, and Grassmann orbitopes. We study their face lattices, their algebraic boundaries, and representations as spectrahedra or projected spectrahedra.
Centrally symmetric polytopes with many faces
 ISRAEL J. MATH
, 2011
"... We present explicit constructions of centrally symmetric polytopes with many faces: (1) we construct a ddimensional centrally symmetric polytope P with about 3d/4 â (1.316) d vertices such that every pair of nonantipodal vertices of P spans an edge of P, (2) for an integer k â¥ 2, we construct ..."
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We present explicit constructions of centrally symmetric polytopes with many faces: (1) we construct a ddimensional centrally symmetric polytope P with about 3d/4 â (1.316) d vertices such that every pair of nonantipodal vertices of P spans an edge of P, (2) for an integer k â¥ 2, we construct a ddimensional centrally symmetric polytope P of an arbitrarily high dimension d and with an arbitrarily large number N of vertices such that for some 0 < Î´k < 1 at least (1 â (Î´k) d) ( N) k ksubsets of the set of verticesspan faces of P, and (3) for an integer k â¥ 2 and Î±> 0, we construct a centrally symmetric polytope Q with an arbitrarily large number of vertices N and of dimension d = k1+o(1) such that at least ( 1âk âÎ±) ( N) ksubsets k of the set of vertices span faces of Q.
NEIGHBORLINESS OF THE SYMMETRIC MOMENT CURVE
, 2011
"... We consider the convex hull Bk of the symmetric moment curve Uk(t) = cost,sint,cos3t,sin3t,...,cos(2k−1)t,sin(2k−1)t ..."
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We consider the convex hull Bk of the symmetric moment curve Uk(t) = cost,sint,cos3t,sin3t,...,cos(2k−1)t,sin(2k−1)t
Applications and Combinatorics in Algebraic Geometry
"... Algebraic Geometry is a deep and wellestablished field within pure mathematics that is increasingly finding applications outside of mathematics. These applications in turn are the source of new questions and challenges for the subject. Many applications flow from and contribute to the more combinat ..."
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Algebraic Geometry is a deep and wellestablished field within pure mathematics that is increasingly finding applications outside of mathematics. These applications in turn are the source of new questions and challenges for the subject. Many applications flow from and contribute to the more combinatorial and computational parts of algebraic geometry, and this often involves realnumber or positivity questions. The scientific development of this area devoted to applications of algebraic geometry is facilitated by the sociological development of administrative structures and meetings, and by the development of human resources through the training and education of younger researchers. One goal of this project is to deepen the dialog between algebraic geometry and its applications. This will be accomplished by supporting the research of Sottile in applications of algebraic geometry and in its applicationfriendly areas of combinatorial and computational algebraic geometry. It will be accomplished in a completely different way by supporting Sottile’s activities as an officer within SIAM and as an organizer of scientific meetings. Yet a third way to accomplish this goal will be through Sottile’s training and mentoring of graduate students, postdocs, and junior collaborators.